Announcements:

• Problems 1, 2, and 5-12 are due on Wednesday (April 3rd)

Homework Assigments

Week of April 24 to 28
Assignments Made on:

Monday: Study: Here are selected homework solutions (Last modified December 31, 1969). Note the time stamp. Please let me know if these have not been updated to what has currently been turned in. As agreed, the assignment ending with problem 153 is due Wednesday. Let's just start out assuming this week's assignment will be due a week from Wednesday and get it over with. Do: Let $\beta\subset \mathcal P(X)$ be a cover of $X$. Show that $\beta$ is a basis for $\tau(\beta)$ if and only if given $U$ and $V$ in $\beta$ and $x\in U\cap V$ there is a $W\in\beta$ such that $x\in W\subset U\cap V$.

Wednesday: Study: Do: If $X$ is a finite dimensional normed space, show that the weak topology is the same as the norm topology. (I suggest using the dual basis.) Show that if $X$ is an infinite dimensional normed space, then every nonempty weakly open set is unbounded. (In addition to showing that the topologies are different, this also implies that $x\mapsto \|x\|$ is not weakly continuous.) I suggest showing that given $\phi_1,\dots,\phi_n\in X^*$, then $\bigcap \ker \phi_i \not=\emptyset$. A topological space $(X,\tau)$ is called Hausdorff if given $x\not=y$ in $X$ there are open neighborhoods $U$ and $V$ of $x$ and $y$, respectively, such that $U\cap V=\emptyset$. Prove that the weak topology on a normed space $X$ is Hausdorff. Not to be turned in: The product toplogy is one of the more ubiquitous objects in elementary topology. Let $(X_a,\tau_a)$ be a topological space for all $a\in A$. Recall that the Cartesian product $\prod_{a\in A}X_a$ is the set of all functions $x:A\to \bigcup_{a\in A}X_a$ such that $x(a)\in X_a$. If $a_0\in A$, then the projection $p_{a_0}$ onto the $a_0$-factor is the map $p_{a_0}:\prod_{a\in A} X_a\to X_{a_0}$ given by $p_{a_0}(x)=x(a_0)$. The product topology on $\prod_{a\in A} X_a$ is the initial topology induced by the projections maps. Thus the product topology is the smallest topology on the product such that each projection is continuous. A subbasis is given by the sets $U(a,V)=p_a^{-1}(V)$ for any $a\in A$ with $V$ open in $X_a$. Let $(x_\lambda)$ be a net in $\prod_{a\in A}X_a$. Then $x_\lambda \to x$ in the product topology if and only if $x_\lambda(a)\to x(a)$ for all $a\in A$. (So the product topology can be thought of as the topology of pointwise convergence.) The Tychonoff Theorem asserts that the (arbitrary) product of compact spaces is compact in the product topology. We'll use this to prove the Alaoglu Theorem in due course. Right now, I want to point out that our result about accumulation points of sequences does not hold in general topological spaces. For each $\alpha\in\ell^\infty$, let $D_\alpha$ be a closed disk in $\mathbf C$ such that $\alpha_n\in D_\alpha$ for all $n\ge 1$. Then $Z=\prod_{\alpha\in \ell^\infty} D_\alpha$ is compact in the product topology. Let $(z_n)\subset Z$ be the sequence given by $z_n(\alpha)=\alpha_n$. Then $(z_n)$ has accumulation points (just because $Z$ is compact), but no converent subsequences.

Friday: Study: This is probably optimistic for Friday. But will be assigned when we get to convexity. Comment: Lecture starts at 10:10. It is unlikely but possible that one day during the term--due to unforseen circumstances--I might be late. So far that hasn't happenned, and if it did I would apologize. I would expect the same from you. Do: Let $S$ be a subset of a vector space $V$. Define $\operatorname{conv}(S)$ to be the collection of sums of the form $\sum_{k=1}^n \lambda_k x_k$ such that $n\ge1$, $x_k\in S$, $\lambda_k\ge0$ and $\sum_{k=1}^n \lambda_k=1$. Show that $\operatorname{conv}(S)$ is the smallest convex subset of $V$ containing $S$. We call $\operatorname{conv}(S)$ the convex hull of $S$. Let $f:(X,\tau)\to (Y,\sigma)$ be a function between topological spaces. Show that $f$ is continuous if and only if $f$ takes convergent nets to convergent nets. That is, $f$ is continuous if and only if given $x_\lambda\to x$ in $X$ we have $f(x_\lambda)\to f(x)$ in $Y$. Let $X$ be a normed vector space. Show that a net $(x_\lambda)$ converges to $x$ weakly if and only if $\phi(x_\lambda)\to \phi(x)$ for all $\phi\in X^*$. Does a weakly convergent net $(x_\lambda)$ have to be bounded? Let $(x_\lambda)$ be a net in a compact space $X$. Show that $(x_\lambda)$ has an accumulation point. I suggest letting $F_{\lambda_0} =\overline{\{\, x_\lambda:\lambda\ge\lambda_0\,\}}$ and looking at $x\in \bigcap_\lambda F_\lambda$. (You should compare this to the corresponding proof in metric spaces. And yes, the converse holds. If every net in $X$ has an accumulation point, then $X$ is compact.) Set $(x_n)$ be a sequence in a metric space $X$. Show that $x$ is an accumulation point of $(x_n)$ if and only if $(x_n)$ has a subsequence converging to $x$. DO NOT WORK YET!

Week of April 29 to May 8
Assignments Made on:

Monday: Study: We agreed that problems 33 to 49 are morally due Wednesday, but legally due Friday. Tychonoff: Here are some proofs of the Tychonoff and Jordan-von Neumann Theorems. Do:

Wednesday: Study: Do:

Wednesday May 8: Study: Here are selected homework solutions (Last modified December 31, 1969). Do: We'll assume we're working is a separable complex Hilbert space from now on. Let $X=\ell^2$. Show that the sequence $(e_n)$ converges weakly to $0$. (As usual, $e_n=\delta_n$. Here is another example to see that the norm is not well behaved with respect to the weak topology.) Let $\bigl(V,(\cdot\mid\cdot)\bigr)$ be pre-inner product space. Let $N=\{\,v\in V:\|v\|=0\,\}$ be the subspace of length zero vectors. Show that $$ (v+N\mid w+N)=(v\mid w)$$ is a well-defined inner product on $V/N$. (Optional) Let $\bigl(V,(\cdot\mid \cdot)\bigr)$ be an inner product space. We want to see that its completion $\widetilde V$ is a Hilbert space without invoking the Jordan-von Neumann Theorem. I suggest the following. Let $i:V\to \widetilde V$ be the natural map. Show that if $(v_{n})$ and $(w_{n})$ are Cauchy in $V$, then $\bigl((v_{n}\mid w_{n})\big)$ is Cauchy in $\mathbf F$. Show that if $i(v_{n})$ and $i(v_{n}')$ converge to $v$ in $\widetilde V$ while $i(w_{n})$ and $i(w_{n}')$ converge to $w$ in $\widetilde V$, then $\lim_{n} (v_{n}\mid w_{n}) = \lim_{n}(v_{n}'\mid w_{n}')$. (Both the limits exist by the previous part.) Show there is a function $(\cdot\mid\cdot)_{0}$ on $\widetilde V\times\widetilde V$ such that $(v\mid w)_{0}=\lim(v_{n}\mid w_{n})$ whenever $i(v_{n})\to v$ and $i(w_{n})\to w$ in $\widetilde V$. Show that $(\cdot\mid\cdot)_{0}$ is an inner product on $\widetilde V$ such that $i:V\to\widetilde V$ is inner product preserving. In particular, $\|v\|=(v\mid v)_{0}^{\frac12}$ for all $v\in \widetilde V$. Let $E$ be a nonempty subset of a Hilbert space $H$. Let $Y$ be the subspace spanned by $E$. Then $E^{\perp\perp}$ is the closure of $Y$ in $H$. (Optional) Some technical niceities. Let $V$ be a complex vector space. Let $V^o$ be the same additive group and $\iota:V\to V^o$ the identity map. Define scalar multiplication on $V^o$ by $\lambda\cdot \iota(v)= \iota(\overline{\lambda}v)$. Then $V^o$ is a complex vector space called the conjugate space to $V$. If $H$ is a Hilbert space, show that $H^o$ is a Hilbert space which is isometrically isomorphic to $H^*$. Let $P\in\mathcal L(H)$ be the orthogonal projection onto a nonzero subspace $W$. Show that $P=P^*=P^2$ and that $\|P\|=1$. Conversely, show that if $P\in\mathcal L(H)$ and $P=P^*=P^2$, then $P$ is the orthogonal projection onto its range. A linear map $V:H\to H$ is called an isometry if $\|V(x)\|=\|x\|$ for all $x\in H$. Show that the following are equivalent. $V$ is an isometry. $\bigl( V(x) \mid V(y)\bigr)=(x\mid y)$ for all $x,y\in H$. $V^*V=I$. A surjective isometry $U:H\to H$ is called a unitary. Show that the following are equivalent for $U\in \mathcal H$. $U$ is a unitary. $U$ is invertible with $U^{-1}=U^*$. If $\{e_n\}$ an orthonormal basis for $H$, then $\{U(e_n)\}$ is an orthonormal basis for $H$. (Remark, (3) imples (1) is not true unless we know $U$ is both linear and bounded.) (Dini's Theorem) Suppose that $X$ is a compact metric space and $(f_n)\subset C(X)$ is such that there is a $f\in C(X)$ such that $f_n(x)\nearrow f(x)$ for all $x\in X$. Show that $f_n\to f$ in $C(X)$. Equivalently, show that $f_n\to f$ uniformly on $X$. (There are probably lots of ways to do this problem. But I found it helpful to note that since the convergence is monotonic, if $f_n\not\to f$ uniformly, then there is a $\epsilon_0>0$ such that $\|f-f_n\|_\infty\ge\epsilon_0$ for all $n$.) Let $T$ be a normal operator. Show that $v$ is an eigenvector for $T$ with eigenvalue $\lambda$ if and only if $v$ is an eigenvector for $T^*$ with eigenvalue $\overline \lambda$. (Optional: More fun from the past) Let $H$ be a finite-dimensional complex Hilbers space. Suppose that $T\in\mathcal L(H)$ is normal. Show that $H$ has an orthonormal basis of eigenvectors for $T$. (Since we're working over $\mathbf C$, we know that $T$ has at least one eigenvector $v$. Let $W=\mathbf C \cdot v$. Agrue that $W^\perp$ is invariant for both $T$ and $T^*$ and that the restriction $T|_{W^\perp}$ of $T$ to $W^\perp$ is a normal operator on $W^\perp$. Now use induction.) (Everything you every wanted to know about partial isometries.) We call $U\in\mathcal L(H)$ a partial isometry if there is a closed subspace $E$ on which $U$ is isometric and $U(E^\perp)=\{0\}$. Suppose $U$ is a partial isometry (on $E$ as above) and $P:=U^*U$. Observe that $$(P(x)\mid x)=\|x\|^2.$$ Conclude that $P(x)=x$ and that $P$ is the orthogonal projection onto $E$. (First use Cauchy-Schwarz to show $\|P(x)\|=\|x\|$.) If $U$ is a partial isometry, then show that $U=UU^*U$. (Hint: $U-UU^*U=U(I-P)$.) Conversely, show that if $V\in\mathcal L(H)$ is such that $V^*V$ is a projection, then $V$ is a partial isometry. Show that if $U$ is a partial isometry, then $U^*$ is a parital isometry with space the range of $U$. Describe the sense in which $U$ and $U^*$ are inverses to each other. (Think of $U$ as a operator from its space $E$ onto its range.) (Optional Diagonalizable Operators) Suppose that $\{ e_n \}_{n=1}^\infty$ is an orthonormal basis in a complex Hilbert space $H$. Let $\alpha\in\ell^\infty$. Show that there is an operator $T\in\mathcal L(H)$ such that $T(e_n)=\alpha_n e_n$ and that $\|T\|=\|\alpha\|_\infty$. (When such a basis exists, we say that $T$ is diagonalizable.) Show that such a $T$ is normal, that $T$ is self-adjoint if and only if each $\alpha_n$ is real, and positive if and only if each $\alpha_n\ge0$. (Optional) Let $H=L^2([0,1])$ and define $T:H\to H$ by $T(h)(x)=xh(x)$. Show that $\|T\|=1$, $T\ge 0$, and that $T$ has no eidenvectors. Hence $T$ is not diagonalizable. (Compare with the finite dimensional case: aka problem #60.) Show that $T$ is compact if and only if $|T|$ is compact. (Recall that if $U|T|$ is the polar deomposition of $T$ then $U^*U$ is the orthogonal projection onto the space of $U$ which is the closure of the range of $|T|$. Use this to show $U^*T=|T|$.) Show that if $I$ is any (not necessarily closed) nonzero ideal in $\mathcal L(H)$ then $\mathcal{L}_f(H)\subset I$. In other words, the finite rank operators are a mininmal ideal in $\mathcal L(H)$. (Show $\mathcal {L}_f$ has nontivial intersection with $I$ and that $\mathcal{L}_f(H)$ has no nontrivial proper ideals.) In lecture, we saw that a linear map $T:H\to H$ whose restriction to the unit ball is weak-norm continuous was a compact operator. Here we want to see that if $T:H\to H$ is weak-norm continuous, then $T$ is a finite-rank operator. (Observe that for such a $T$, $x\mapsto \|T(x)\|$ is weakly continuous. Hence there are $x_1,\dots,x_n\in H$ such that $|(x\mid x_k)|<1$ for all $k$ implies that $\|T(x)\|<1$.) Show that a norm-weak continuous map is actually norm-norm continuous. (Optional) Let's review the final details of the Spectral Theorem for Normal Compact Operators. Let $T$ be a normal compact operator on $H$. Let $\alpha=\{e_k\}$ be a set of orthonormal eigenvectors for $T$, and let $P$ be the orthogonal projection onto $E=\overline{\operatorname{span}\{e_k\}}$. Show that $P$ and $T$ commute. Let $S=(I-P)T$. Show that $S$ is compact and normal. If $S=0$, then show that any unit vector $e_0\in E^\perp$ is an eigenvector for $T$. If $S\not=0$, then our partially proved lemma implies that $S$ has en eigenvector $e_0$ such that $S(e_0)=\lambda e_0$ with $|\lambda|=\|S\|$. Show that $e_0 \in E^\perp$ and that $e_0$ is an eigenvector for $T$. Conclude that $\alpha\cup\{e_0\}$ is an orthonormal set of eigenvectors for $T$.

Thursday, May 16: STATUS: The above homework assignments complete the "required" part of the course. They'll be due sometime before the end of term. Solutions: Here are selected homework solutions (Last modified December 31, 1969) up through assignment 49. The Spectral Theorem: We will be following sections 2-4 of my notes on the Spectral Theorem from my web page. A slightly different, but perhaps more friendly version, can be found in Chapters 1 and 2 of my C*-algebra notes from my web page. If you do puruse either of these, I would appreciate knowing about any typos, thinkos, or exposition that seems opaque or could be improved. Let $\Omega$ be a connected open subset of $\mathbf C$. (We call $\Omega$ a domain.) If $A$ is a Banach algebra, then we call $f:\Omega\to A$ strongly holomorphic if $f'(z):=\lim_{h\to 0}h^{-1}(f(z+h)-f(z))$ exists (in $A$) for all $z\in \Omega$. Observe that if $f$ is strongly holomorphic and $\phi\in A^*$, then $\phi\circ f$ is holomorphic on $\Omega$ in the usual sense. Show that if $f$ is strongly holomorphic on $\mathbf C$ and bounded, then $f$ is constant. Suppose that $\Omega$ is domain. Let $H(\Omega)$ be the collection of complex valued holomorphic functions on $\Omega$. The Deformation Invariance Theorem implies that if $\phi\in H(\Omega)$ and $\Gamma_0$ and $\Gamma_1$ are homotopic closed contours in $\Omega$, then $$\int_{\Gamma_0} \phi(z)\,dz = \int_{\Gamma_1} \phi(z)\,dz.$$ Use this result to show that the same conclusion holds if $\phi$ is replaced by a strongly holomorpic $A$-valued function $f$ on $\Omega$ for a Banach algebra $A$. Let $A$ be the subset of $2\times 2$ complex matrices given by $$\Bigl\{ \begin{pmatrix}a & b \\ 0 & a \end{pmatrix}:a,b\in\mathbf{C}\,\Bigr\}.$$ Observe that $A$ is a two dimensional unital commutative Banach algebra and that $\operatorname{Rad}(A)\not=\{0\}$. (Hint: any complex homomorphism on $A$ is in particular a linear map from $A$ to $\mathbf C$.) Homework Solutions:Here are the final selected homework solutions (Last modified December 31, 1969).

Dana P. Williams
Last updated May 28, 2019 09:25:12 EDT